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%I #45 Jun 13 2024 14:57:52
%S 1,6,3,27,30,6,108,205,90,10,405,1188,870,210,15,1458,6279,6888,2730,
%T 420,21,5103,31306,48622,28308,7070,756,28,17496,149985,318726,256914,
%U 92988,16002,1260,36,59049,698256,1984950,2136150,1054305,260316,32760,1980,45,196830
%N T(n, k) is the total number of non-symmetric peaks in all partitions of n with exactly k blocks, n >= 4, 3 <= k <= n-1.
%H W. Asakly and Noor Kezil, <a href="https://arxiv.org/abs/2401.01687">Counting symmetric and non-symmetric peaks in a set partition</a>, arXiv:2401.01687 [math.CO], 2024.
%F T(n,k) = binomial(k-1, 2) * Stirling2(n-1, k) + 2 * Sum_{j=3..k} binomial(j, 3) * Sum_{i=3..n-k} j^(i-3) * Stirling2(n-i, k).
%e The triangle T(n, k) begins:
%e 4| 1
%e 5| 6 3
%e 6| 27 30 6
%e 7| 108 205 90 10
%e 8| 405 1188 870 210 15
%e 9| 1458 6279 6888 2730 420 21
%e 10| 5103 31306 48622 28308 7070 756 28
%e .
%e T(5,3) represents the partitions of the set {1,2,3,4,5} into 3 blocks:
%e The canonical form of a set partition of [n] is an n-tuple indicating the block in which each integer occurs. The non-symmetric peaks in the canonical sequential form are listed:
%e (1, 2, 3, 1, 1) -> 1 non-symmetric peak, (2, 3, 1)
%e (1, 2, 3, 1, 2) -> 1 non-symmetric peak, (2, 3, 1)
%e (1, 2, 3, 1, 3) -> 1 non-symmetric peak, (2, 3, 1)
%e (1, 2, 2, 3, 1) -> 1 non-symmetric peak, (2, 3, 1)
%e (1, 1, 2, 3, 1) -> 1 non-symmetric peak, (2, 3, 1)
%e (1, 2, 1, 3, 2) -> 1 non-symmetric peak, (1, 3, 2)
%p T := (n, k) -> binomial(k-1, 2) * Stirling2(n-1, k) + 2 * add(binomial(j, 3) * add(j^(i-3) * Stirling2(n-i, k), i=3..n-k), j = 3..k): seq(print(seq(T(n, k), k = 3..n-1)), n = 4..10);
%t T[n_, k_] := Binomial[k-1, 2] * StirlingS2[n-1, k] + 2 * Sum[Binomial[j, 3] * Sum[j^(i-3) * StirlingS2[n-i, k], {i, 3, n-k}], {j, 3, k}];Table[T[n, k], {n, 4, 12}, {k, 3, n-1}]
%o (PARI) T(n, k) = binomial(k-1, 2) * stirling(n-1, k, 2) + 2 * sum(j=3, k, binomial(j, 3) * sum(i=3, n-k, j^(i-3) * stirling(n-i, k, 2)));
%Y Cf. A008277 (Stirling2).
%Y Cf. A373288.
%Y Cf. A027471 (1st column), A033487 (subdiagonal).
%K nonn,tabl
%O 4,2
%A _Noor Kezil_, Jun 07 2024
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